NOTES ON 



Elementary Mechanical Drawing 



COURSE I, 



UNIVERSITY OF MICHIGAN 

1905 



GEORGE WAHR 

PUBI,ISHER A^fD BOOKSBHER 
ANN ARBOR, MICH. 



NOTES ON 



Elementary Mechanical Drawing 



COURSE I. 



UNIVERSITY OF MICHIGAN 

1905 



GEORGE WAHR 
Publisher and Bookseller 

ANN ARBOR, MICH. 



•,0pl83 rtOCtttVtJU 

. MAR 6 1905 

COPY B. 






Copyright, 1905, 

BY 

George Wahk. 



PENCILING. 

In the construction of the following- problems 
familiarit}^ with the instruments and accuracy in 
execution are the chief objects to be attained. 

Before commencing- au}^ drawing the student 
should see that his instruments are in working 
order. 

For mechanical drawing the pencil should be as 
hard as ma}^ be and still give a distinct line with- 
out creasing the paper. The lead must be perfectly 
smooth, having no grit or foreign substance, and 
tough enough to bear sharpening to a fine edge or 
point without breaking. 

For freehand drawing pencils of various degrees 
of hardness are required for shading and outline 
work. 

For right line work the pencils should be sharp- 
ened to an edge like a chisel, and the corners 
slight}" rubbed off. This gives a point that wears 
much longer than the round one and is not so easil}" 
broken. The wood is cut awa}" with a knife and 
the lead ground down upon a piece of fine emer)^ 
paper. 

In case the pencil drawing is to be inked and the 
work is to be made a finished production, the pencil 
should be a HHHHHH or one of equal hardness. 
When the pencil drawing is not to be inked but a 



4 ELKMENTAKY MECHANICAL DKAWING. 

tracing- is to be made from it, a softer pencil of the 
g-rade HHHH should be used. 

Care must be exercised in the use of the hard 
pencil, for if too much pressure is broug-ht to bear 
upon it a crease is made in the paper which remains 
after the lead has been removed b}^ the eraser. 

In order to become an accomplished and reliable 
draftsman, it is absolutely necessar}^ for the student 
to learn from the start to attain the accuracy in pen- 
cil construction which is desired in the iinished 
drawing-, having- ever}^ line in its proper form, size, 
position and relation to every other line. 

All corrections should be made in the pencil 
drawing- and not left to be made during- the process 
of inking- in. 

All lines are drawn from left to rig-ht and should 
be drawn lig-htl3^ 

To draw a line between two points, place the 
pencil upon one of the points, bring- the T square 
or triang-le ag-ainst it and at a distance from the 
other point equal to the width of the pencil point. 
Now move the pencil to the second point, using- the 
straig-ht-edg-e for a g-uide and keeping- the pencil 
always parallel to its first position. 

If it becomes necessary to erase a portion of the 
work, the particles of rubber left upon the paper 
should be removed with a clean piece of linen rag- 
before attempting- to ink the drawing^. 



INKING, 

The pencil drawing- completed, the next step is 
to examine the ruling- or rig-ht line pen, and, if nec- 
essar}^ to place it in perfect working order prepara- 
tor}^ to inking. 

It is desirable for the student using a ruling pen 
for the first time to draw a number of lines at ran- 
dom on a loose sheet of paper using the triangles as 
guides, until he is able to draw them straight. At 
first thought this may seem eas}^ but to one unac- 
customed to the use of the pen the drawing- of a 
straight line will require much practice. 

The lines are drawn by moving the pen from 
left to right, keeping it parallel to the first position, 
for if a lateral movement be given while tracing a 
line, the point of the pen v/ill approach or recede 
from the guiding- edge and the resulting line will 
be wav)^ 

In drawing lines with a right line pen, see that 
both blades bear with equal force upon the paper, 
being pressed lightl}" if the paper is smooth and 
with increasing pressure according to its Toughness. 

The pen should be inclined slightl}^ in the direc- 
tion of motion, and but sufficient pressure to guide 
it should be g-iven against the straight edge, other- 
wise the blades will be forced together and a line of 
unequal width will be the result. 



6 KLKMKNTARY MECHANICAL DRAWING. 

To lessen the dang-er of blotting:, the g-uiding- 
edg-e should be slig-htl}' removed from the line to be 
drawn so that the pen point in tracing- the line will 
not come in contact with it. 

Where several lines radiate from a point the 
lines should be drawn from the point, not toward 
it, allowing" each line to dr}^ before drawing* the 
succeeding- one in order to prevent a blot which is 
ver}^ likel}" to be made at the point. 

As in the case of the rig-ht-line pen, the blades 
of the compass pen must bear evenl}^ upon the 
paper, and to attain this the compass legs will have 
to be adjusted in the joints so that the}^ will be per- 
pendicular to the paper. 

In describing- arcs, allow onl}^ the weig-ht of the 
compasses to bear upon the needle point, while a 
slig-ht pressure ma^^be g-iven to the pen point, varv- 
ing- according- to the surface of the paper. 

The top of the compasses can be slig-htly in- 
clined in the direction of the motion. 

It is alwa3^s better to describe the smallest of a 
number of concentric circles first. 

Since it is easier to make a rig-ht line meet a 
curve than the reverse, do the compass work before 
the rig-ht-line work. 

In erasing- ink lines it is far preferable to use the 
steel eraser. After erasing- rub the surface with a 
clean, smooth, hard substance to prevent the ink 
from spreading-. The fing-er-nail does this verv well. 

The student should always strive to be neat, and 



INKING. 




should keep covered that portion of the drawing- 
which he is not working- upon at the time. 

Fig-. 1 (a). To lay off on a given arc a length 
equal to a given right line. Let BE be the arc and 
AB the g-iven line. At an}^ point of the arc, as B, 

draw a tang-ent BF. On r /i g b 

this tang-ent la^^ off BA 

equal to the g-iven line. 

Divide this leng-th into 

four equal parts. From 

G, the point of division 

nearest to B, as center 

strike an arc through A ^^^' ^ ^^)- 

cutting- the arc BE in D. BD is the arc required. 

If the line AB, when laid off on the arc, g-ives a 
distance which subtends an ang-le g-reater than 60° 
divide it into a number of equal parts such that 
each part when laid off on the arc would give a dis- 
tance that would subtend an ang-le of 60° or less. 
Then step this distance off on the arc as many times 
as the line AB was divided into parts. 

(b) To rectify the arc 
of a circle. Let AB be the 
g-iven arc, and C the cen- 
ter. At either extremity 
of the arc, as B, draw the 
tang-ent BE. Bisect the 
chord AB in H and pro- 
duce it to I, making- BI 
equal to BH. From I as fig. i (b). 



I. 




8 EI.KMKNTARY MECHANICAL DRAWING. 

center strike an arc throug-h A cutting" BE in D. 
BD is the length of the given arc. 

If the arc AB subtends an angle greater than 
60° divide it into a number of equal parts such that 
each part would subtend an angle of 60° or less. 
Rectif}^ one of these parts and step the rectified 
length off on the tangent as man}^ times as there 
are divisions in the arc. 

Fig". 2. To divide a given line into any mnnber 
of equal farts. Let AB be the given line and 6 the 

number of parts. Draw ^^ 

^ ^ ^-^^t::^ — ^ — ^^ — ^^ — ^ — kQ 

an}" line AC making- a con- ^ ~^V J^^ "^x ^\ ^\ 

venient ansfle with AB. "V ^. \ 



te^^ vv ^y.^ ^^M^. - ^\ 



\ 



On AC lay off six equal "^c 

spaces of any leng-th. Draw 

BC and throug-h the points of division on AC pass 
parallels to BC. The intersections of these parallels 
with AB g-ives the required points of division. 

Fig-. 3. To construct a reg- 
ular pentagon upon a given side. 
Let AB be the given line. Bi- 
sect it at G b}^ the perpendicu- 
lar GD. At B erect the perpen- 
dicular BF equal to the g-iven 
line. From G as center with 
radius GF strike an arc FH in- 
tersecting- AB produced in H. 
From A and B as centers with radius equal to AH 
describe arcs intersecting at D. With A and D as 
centers and radius AB strike arcs intersecting- at E. 




geomp:tricai. constructions. 



9 



"With B and D as centers and radius AB strike arcs 
intersecting- at C. The points A, B, C, D, and E 
are the vertices of the required pentagon. 

To construct a regulai^ decagon upon a given side. 
Let AB be the given side. Find point D as above. 
From D as center strike a circumference through A 
and B. Step oii the chord AB upon this circum- 
ference closing- upon A. The points thus found are 
the vertices of the required decag-on. 

Fig-. 4. To inscribe a regular 
pentagon in a circle. Let ADBC 
be the g"iven circle. Draw an}^ 
two perpendicular diameters, AB 
and CD. Bisect the radius JB at 
I. From I as center v/ith radius 
IC strike an arc intersecting- AB 
in K. From C as center with CK 
as a radius strike an arc cutting- the circle in F. 
CF is one side of the required pentag*on. 

Fig. 5. To construct an 
ellipse,, having given the 
principal axes, (Method 
here described is illus- 
trated b}^ left half of Fig. 
5 (a) and is used to con- 
struct onl}^ the upper half 
of Fig-. 5). Let AB and CD be the given axes. 
Through the points A, B, C, D, draw the rectangle 
EFGH. Divide AE into au}^ number of equal parts, 
and AO into the same number. Number the points 





Fig. 5 (a). 



10 



KI.KMENTAKY .AIKCHANICAL DK AWING. 




of division from A toward E and from A toward O. 
From C draw a line to any point, as 4, on AK and 
from D a line through 4 on AO. The intersection 
of these lines is a point on the curve. Other points 
are found in the same manner. 

(Method here described 
is illustrated by left half 
of Fig*. 5(b) and is used to 
construct only the lower 
half of Fig-. 5). Let AB 
and CD be the axes. Draw 
the rectangle EFGH as 
above. Draw AC and AD. From B draw any line 
B^^"* cutting- AC in s ^md from s a parallel to CD cut- 
ting- AO at s ' Draw Dj' intersecting- Bj at 3' , 
This is a point of the curve. Other points are 
similarly found. 

Fig-. 6. To find a circle 
zvhich shali cii'ctijnscrihe a 
given number of circles of a 
given size^ the inscribed cir- 
cles to be tangent to each other 
and to the circumscribing cir- 
cle. 

Let the number of circles 
be twelve and let the diam- 
eter be AB as shown in the fig-ure. Upon this diam- 
eter as a side construct the equilateral triang-le ABC. 
Draw CD bisecting- AB. With C as center and CB 
as radius strike an arc cuttinof CD in D. With D as 




GEOMETRICAL CONSTKUCTIONS. 



11 



center and DE == DA + /^ AB as radius describe a 
circle, which is the required circumscribing" circle. 
With D as center and DA as radius describe a circle, 
in which inscribe a reg-ular dodecagon using- AB as 
one side. The vertices of this pol3^gon are the cen- 
ters of the twelve inscribed circles. 

Fig. 7. To construct an ellipse, haviiig given the 
^rincii>al axes. Let AB and CD be the given axes. 
From C or D as center with radius equal to O A strike 
arcs cutting AB at F 
and F'. These are the 
foci of the curve. On 
FO take any points of 
division, as 7,2, j,^, etc. 
Point I should be ver}^ 
near F and the spacing 
between points ma}^ 
increase toward O as 
shown. From F as cen- 
ter with radius equal to 
Bj strike the arc of a circle near B. A similar arc 
ma3^ be struck near A with the same radius and F' 
as center. From F' with a radius equal to A/ cut 
the first drawn arc in a and a which are two points 
of the required ellipse. From F with the same 
radius cut the arc previously described about F' as 
center with radius B/, thus obtaining- two more 
points of the required curve. In like manner with 
the radii B^ and K2 find the points h and h' and also 
the corresponding- points at the other extremity of 




Fig. 7. 



12 



KI.EMENTAKY MECHANICAI, DRAWING. 



the axis. The other points on the curve are found 
by the same method. Then with the irreg-ular 
curve strike a fair curve throu^^h the points. This 
solution depends upon the principle that the sum of 
the distances of an_v point on the curve from the 
foci is a constant. 

Where g'reat accuracy is not desired, the same 
principle ma^^ be used b3^ fastening* a thread of 
length AB to pins at F and F', and drawing- a pen- 
cil along- the loop thus formed, keeping- the thread 
alwa3^s taut. 

To draw a tangent at any point on an ellipse. Let 
ACBD be the ellipse and T the g*iven point. Draw 
TF and TF', producing the latter to E. The bisec- 
tor of ang-le ETF is the required tangent. 

To draw a tangent to an ellii>se from a j)oint out- 
side. Let ABCD be the ellipse and P the given 
point. From P as center strike an arc throug-h F', 
and from F as center with a radius equal to AB cut 
the first arc in G and H. Lines drawn from P 
bisecting- arcsF'H and F'G are the required tangents. 

Fig. 8. To con- 
striict an hyferl)ola, 
having given the ver- 
tices and the foci. Let 
A and B be the given 
vertices and F and F' 
the g-iven foci. Take 
any points on AB be- 
yond F', as i,2,s, etc. fig. 8. 




GKOMKTRICAI, CONSTRUCTIONS. 



13 



From F and F' as centers with radius equal to A/ 
strike the arcs ah and a b' , From the same centers 
with a radius B/ cut the first arcs in a, <5, a! ^ and b' , 
In the same manner find the points c^ d, c and d' , 
The points thus located determine the curve. 

To draw a tangent to an hyperbola at a given 
point of the cui^ve. Let T be the g-iven point. Join 
this point with the foci b}^ the lines TF and TF'. 
Bisect the angles between these lines by the line TR 
which is the required tangent. 

To draw a tangent to an hyperbola from a point 
outside the curve. Let P be the given point. From 
P as center strike an arc throug-h F'. From F as 
center with radius equal to AB cut this arc in C. 
Bisect the arc F'C by the line PM. This is the 
required tangent. 

Fig. 9. To construct a parabola upon a given 
axis and base. Let EG be the given base and MC 
the given axis. Take the vertex A upon MC, bisect 
EB in D, draw AD and per- 
pendicular to it the line DC 
cutting- MC in C. Lay off 
from A the distances AM 
and AF equal to BC. At M 
erect the perpendicular IJ to 
MC. F is the focus and IJ 
the directrix of the curve. 
Divide the line AB in an}^ points, as /, 2, j, etc., and 
through these points erect perpendiculars to AB. 
From F as center with radius equal to M/ strike 




14 



KI.KMENTARY MECHANICAL DRAWING. 



arcs cutting- the perpendicular throug-h /; from F 
as center with radius equal to M2 cut the perpen- 
dicular through 2, Proceed in this manner, finding- 
the points of intersection which are points of the 
required curve. This construction depends upon 
the principle that a parabola is the path of a point 
moving- so that its distance from a point K is alwa3^s 
equal to its distance from a g-iven line IJ. 

To draw a tangent to a farahola at a given point 
on the curve. Let T be the given point. Draw T'N 
perpendicular to IJ and T'F to the focus. Bisect 
the angle between these lines b}^ the line T'R, 
which is the required tangent. 

To draw a tangent to a parabola from a point 
outside the curve. Let P be the g-iven point. From 
P as center with a radius PF strike an arc FH cut- 
ting- IJ in H. Bisect this arc bv the line TP, which 
is the required tang-ent. 

Fig-. 10. To construct 
an ellipse, having given 
the principal axes. Let 
AB and CD be the g-iven 
axes. Upon each of these 
as diameter describe a cir- 
cle. From the center draw 
any radius, as Oj, cutting- 
the circles at s and 3 , 
Throug-h 3 draw a line 
parallel to AB and through 
3 a line parallel to CD, g-iving- at their intersection 




Fig. 10. 



GEOMETRICAL CONSTRUCTIONS, 



15 










Fig. 11. 



s'\ a point on the curve. Other points are found in 
the same manner. 

Fi^. 11. To construct the spiral of Archi7nedes, 
Let O be the fixed point and Oa the distance which 
the point recedes from O in ^-z-^^ 

making- one complete revo- 
lution. Prom O as center 
describe the circumference 
of a circle of an}' convenient 
radius and divide this cir- 
cumference and the line Oa 
into the same number of equal parts. Through the 
points of division on the circle draw radii to O and 
through the points of division on Oa strike circles 
from the center O. The circle from / cuts the 
radius O/' at /', the circle from 2 cuts the radius 
O2' at 2', and so on, giving- the points a^ i\ 2, etc., 
which determine the curve. 

Fig. 12. To construct the Logarith^nic spiral. 
Let O be the fixed point and a a point on the curve. 
Draw Oa, and b}^ means of an aux- 
iliary^ circle draw the radiating lines 
Oi , O2 , O3 , etc. , throug-h O. These 
lines must be separated b}^ equal an- 
g-les. From a let fall a perpendicular 
upon O/', cutting it at /. From / 
let fall a perpendicular upon Oa cut- 
ting the latter at 2. In the same 
manner b^^perpendiculars alternately 
upon Oi and Oa find the points j, 4, 5, 




Fig. 12. 



etc. PVom 



16 KLKMKNTAKY MKCHANICAI. DRAWING. 

O as center a circle throug-h / cuts the line O/', a 
circle from 2 ctits 02, one from s cuts Oj', etc., 
g-iving- points of the curve. 

Fig". 13. To construct an external epicycloid. Let 
AB be the base and acfi the rolling- circle. Let A 
be the point from which the curve spring's. Divide 




Fig. 13. 

the circumference of the rolling- circle into au}^ num- 
ber of equal parts. Draw a common tangent to the 
two circles at 6\ V>\ (//) Fig-. 1 lav off on the tan- 
gent a length equal to a6' and then by (/O I'^ig. 1 
la}^ off on the base circle the length thus found giv- 
ing a length of arc 5'd' which is equal to arc a6' , 
On the base circle from A to B la}^ off as mau}^ divi- 
sions as are on the rolling circle and make them 
equal in length to arc s' 6' thus giving points /', 2, 
s\ etc. AB is therefore equal to the circumference 
(fcfi. Through the center of the rolling- circle strike 
an arc CD parallel to the base. Draw lines from the 
center of the base circle through the points of divi- 



GKOMKTRICAI. COKSTRUCTIONS. 



17 



sion7',^',j',etc., cutting CD in /,^,j,^,etc. Through 
the points of division on the rolling circle strike 
parallels to the base. From / as center with radius 
equal to that of the rolling" circle strike an arc cut- 
ting the parallel through a at /; and from 2 as cen- 
ter with the same radius cut the parallel through 
b in m. In the same manner find the other points 
of the curve. The same points ma}' be found by 
striking from /' as center, with radius equal to 
chord a6\ an arc cutting the parallel through a in /, 
and from 2 as center w4th radius equal to the chord 
b6' and arc cutting the parallel through b in m. 
Other points are found in a similar manner. 

To draw a tangent at a given point of aji eticy- 
chid. Let r be the given point. Through r draw 
a parallel to the base cutting- the rolling circle in 11, 
From r as center with a radius equal to the chord 
h6' strike an arc cutting* the base in 8' , Draw the 
line 87' and through r perpendicular to this line the 
tangent. 

Fig. 14. To construct a cycloid. Let AB be the 
base and acfi the rolling circle. Let A be the point 
from w^hich the C3xloid springs. La}^ off upon the 




18 ELEMENTARY MECHANICAL DRAWING. 

circle the equi-distant pointvS a, h, r, d^ c, etc., and 
tbroug-h these points draw parallels to the base. 
From A la}^ off AB equal to the circumference of 
the rolling- circle and divide it into as many equal 
parts as the rolling- circle b}^ the points /, 2, j, etc. 
Use the method of Fig. 1, (/;) to obtain the leng-th 
5-d equal to a6. All the divisions on AB are equal 
to the division ^-6. Through the center of the cir- 
cle draw the line CD parallel to the base line and 
erect at A the line AC perpendicular to AB. From 
A, /, 2, J, etc., erect perpendiculars to AB cutting- 
CD in points C, /', ^', s\ etc. From /' as center with 
radius equal to that of the rolling- circle describe an 
arc // cutting the parallel through a at /. From 2 
as center with the same radius cut the parallel 
through b at vi, and with s S-S center and the same 
radius cut the parallel through c at ;/. Continue in 
this manner, finding- the points o, p, f, q, etc. The 
points /, 711, 11^ etc., determine the C3xloid. The same 
points ma}^ be found b}^ letting fall from /', 2', etc., 
perpendiculars upon the base to /, 2, j, etc. From 
I as center with a radius equal to the chord of one 
division on the circle, as a6^ cut the parallel through 
a in /. From 2 as center with radius equal to the 
chord of two divisions of the circle, as b6 cut the 
parallel through h in ;;/. Other points are found in 
a similar manner. 

To draw a tangent at any point on a cycloid, Let 
o be the point. Through o draw a parallel to the 
base cutting the g-enerating circle in d. From o as 



GEOMETRICAL CO^:STRUCTlo:^sS. 



19 




center with radius equal to the distance d6 cut the 
base in 4. Draw 04 and throug-h o perpendicular to 
04 the tangent. 

Fig". 15. To construct a hypo cycloid. Let Ad' B 
be the base and acfi the rolling circle. Let A be 
the point from which the curve springs. La}^ off 
on the base an arc 
AB equal in length ^ ^ 

to the circumfer- 
ence of the rolling 
circle and divide 
this arc and the 
rolling* circle into 
the same number 

of equal parts. Methods {b) and {a) Fig. 1 are 
necessar}^ to accomplish this. 

Through the center of the rolling circle draw 
CD parallel to the base, and throug-h the points of 
division /', 2, j\ etc., on AB, draw radii to AB cut- 
ting CD in 7, 2, J, etc. Through the points of divi- 
sion a, h, c, on the rolling circle strike parallels to 
the base. From / as center with the radius equal 
to that of the rolling circle strike an arc cutting the 
parallel through a in /; from 2 as center with the 
same radius cut the parallel through h in m. In 
like manner find other points. The same point ma}^ 
be found b}^ striking an arc from i' with radius 
equal to the chord a6' cutting the parallel through 
a at /, and from 2 an arc with radius equal to b6, 
cutting- the parallel through h in 7n. 



20 



ELHMKNTAKY MHCHANICAI. DRAWING. 



Note: — Two rolling circles the sum of whose 
diameters is equal to the diameter of the base circle 
describe the same hypoc3xloid. When the diameter 
of the rolling- circle is one-half that of the base cir- 
cle the h ypocycloid becomes a diameter of the base 
circle. 

To drazc a tangent at a given jyoint of a Jiy^o- 
cycloid. Let o be the g-iven point. Throug-h o draw 
a parallel to the base cutting- the rolling- circle in d. 
Prom o as center with a radius equal to the chord 
d6' strike an arc cutting- the base in 4 , Draw the 
line 04 and through o perpendicular to 04 the 
tang-ent. 

Fig-. 16. To construct the involute of a circle. Let 
/, 2, J, -/, 5, 6^ be the g-iven circle and O its center. 
Take any points /, 2, j, etc., 
upon the circumference and let 
/ be the point from which the 
curve springs, and from which 
the leng-ths of the arcs are meas- 
ured. At 2, 7, etc., draw tan- 
g-ents to the circle toward /. 
On the tang-ent at 2 \^y off a 
distance 2-2 equal to the arc 
1-2 by au}^ convenient method, 
as that of {a) in Fig". 1. In 
like manner lay off on the tang-ent at 3 a leng-th j-^^-*' 
equal to the arc i-s and so continue, finding- a suffi- 
cient number of points to locate the curve. The 
curve passes throug-h /, 2, f, 4 ^ etc. 




Fig. 16. 




GKOMETRICAL CONSTRUCTIONS. 21 

Fig-. 17. To construct an internal epicycloid. Let 
the circle K2 Bd' with center at O be the base and 
the circle acfi^iih. center at C be the rolling- circle. 
Divide the circumference of the rolling circle into 
an}^ number of equal arcs— here twelve of length Ka, 

Step off on the base circle as ^ 

man}^ arcs of the same length A^^^^^'^'^^^ii^'^^^ 

as arc A^. The equal arcs ■/( A-'^^^^^^cXf^^ 
are obtained bjr methods (^) 1 \^ '^2 ca ^ :>ii' Ai 
and (tz) of Fig. 1. Since the 
rolling circle in this case is 
larger than the base circle 
the circumference of the 

c -1. -. Fig. 17. 

former will go around the 

latter more than once and points 8\, g\ io\ etc., will 
not fall on A, /', 2, s\ etc., as the}- do in Fig. 17 
which illustrates a special case. From the points 
of division /', 2\ s\ etc., upon the base circle draw 
lines through O and produce them be3-ond O to the 
circle through C, struck from the center O. Num- 
ber the points thus found, 7, 2, j, etc. From O as 
center through the points of division, a, h^ c, etc., 
on the rolling circle strike parallels to the base. 
From I as center vvdth a radius equal to that of the 
rolling- circle cut the parallel through a in /; from 2 
as center with the same radius cut the parallel 
throug-h b in ;//, and so on, finding the points on the 
curve. The same points ma}- be found b^^ striking 
from /' as center with a radius equal to the chord 
aK an arc cutting the parallel through a in /, and 



22 



EI.BMENTAKY MPXHANiCAL DK AWING. 



from 2 with radius equal to the chord /;A an arc 
cutting- the parallel through b in m. 

After passing the point q it will be noticed that 
in striking- the arc about 7 as a center with a radius 
equal to the rolling- circle two points of intersection 
are found with the circle struck throug-h e whose 
center is O. The point which lies on the desired 
curve will be that one nearest q, the point pre- 
viously located. A like choice of points should be 
made where two intersections are obtained. 

To draw a tangent at a given point on the internal 
epicycloid. Let // be the g-iven point. Through ;/ 
strike a parallel to the base cutting the rolling cir- 
cle at c and from ;/ as center with radius equal to 
the chord Kc cut the base inj'. Draw the right 
line s n and throug-h ;/ perpendicular to this line the 
tang-ent. The normal to the curve at ;/ is ns , 

Fig-. 18. To construct the pro- 
jections of the helix. Let a I' be 
the axis and Oa the distance of 
the tracing- point from the axis. 
Then will the projection of the 
axis upon its normal plane be 
found at O, and the helix will be 
projected in the circle adgj. Di- 
vide the circle into au}^ number of 
equal parts, and let ah' be the 
distance which the tracing- point 
moves in the direction of the axis 
w^hile it describes an anerle about fjg. is. 




GEOMETRICAL CONSTRUCTIONS. 23 

the axis subtended b}^ one of the divisions of the 
circle. La}^ off upon the axis a number of divisions 
equal to a b' and throug-h these points of division 
draw perpendiculars to a I' , Throug-h the points 
of division on the circle draw parallels to at cut- 
ting these perpendiculars. If, now, the tracing- 
point is seen at a and upon the line through a! ^ 
when it has traveled to b it will be found upon the 
line through b\ and will therefore be at the inter- 
section of the lines through b and b\. In like man- 
ner it wnll pass through the intersections of the 
lines through c and c , etc. A fair curve drawn 
through these points is the projection upon the 
plane parallel to the axis. 

To drazv a tangent to a helix, Let m be the point, 
which is shown in the projection on the normal 
plane at/. Through au}^ point of the helix below 
7n draw a horizontal line, as at nb\ Through the 
intersection // draw a parallel to at cutting the cir- 
cle in b. At y draw a tangent to the circle and lay 
off upon it a lengthyb equal to the diVcfb, Through 
o draw a parallel to at cutting" nb in ^. Draw^/;^, 
which is the required tangent in the projection upon 
the plane parallel to the axis, as is ^y in the pro- 
jection upon the normal plane. 



LINE SHADING. 

The work on line shading- is illustrated ])y Fi^rs. 
19, 20, 21 and 22, and will be explained by the in- 
structor. 





I.INK SHADING. 



25 




Fig. 22. 

RIGHT LINE HATCHING. 

Sections of objects are made to show hidden 
parts and forms which cannot be well represented 
in plan, front or side elevation. Where a section 
of an object is made, it is usuall}^ hatched, the st3^1e 
of hatching- indicating- the material of which the 
object is composed. The rectangles in Fig. 23 are 
hatched according to the conventional method. 




CAST 




w'ooo 




\JR OUGHT 
IRON 






STLE.L 



BRASS 




S TON I 



Fig. 23. 



TEETH OF GEARS. 

Ill connecting- the moving" parts of machines, 
whether the motion be rotar}^ or rectilinear, where 
an absolutely invariable velocity ratio is desired the 
connection is most frequently made b}^ means of 
g"ears. In the past these g^ears have usualh^ been 
built bv using hard wood teeth in frames of wood 
or iron, or have been cast in iron of the desired form. 
At the present time wooden gears have passed out 
of date and cast g^ears, that is, those in which the 
finished form is taken from the mold, are onl}^ used 
in cheap machiner3% where the speed is slow and 
economy of power transmission is a secondar}^ con- 
sideration, and in an occasional example of a gear 
of ver}^ larg-e size, which cannot readih^ be handled 
for milling- or planing. The process of cutting 
g-ears has been so cheapened that 
in all machine work the cut g-ears 
are used, there being great econ- 
om3^ in power transmission and 
also in weight, since the more per- 
fect action of the cut gears gives 
a large increase in their working- 
strength and consequently permits 
the use of far lighter gears than 
could be used in the cast form. 

If two cylinders, a and /;, Fig. 
24, having fixed parallel axes are fig. 24. 




TKETH OF GKARvS. 



27 



placed so as to be in contact, and if one, as a be set 
in rotation, the other will also rotate, and the}^ will 
have a rolling" contact. Their angular velocity ratio 
will be inversely as their radii, and will remain con- 
stant so long- as the surface friction at their line of 
contact is sufficient to overcome the resistance offered 
b}^ h due to friction and the work it is performing. 
If the load on h is increased to a sufficient amount a 
will slip over ^ to a g-reater or less extent and the 
velocit}^ ratio will not be constant. 

It is to avoid this slipping- that teeth are placed 
upon the rims of the c^^linders to eng-r^ge or mesh 
with each other. It is evident that in order to 
secure a uniform velocit}^ ratio the teeth must have 
a certain definite form, and it is with the determin- 
ation of this form that we are concerned. In order 
to take up the discussion of gear teeth the follow- 
ing- definitions of terms are necessary. Fig-. 25 
shows an end view of a portion of two gears in 




28 ELEMKNTAKY MECHANICAI. DRAWING. 

action. The two cylinders show in the two circles 
ah and cd. These cylinders are pitch cylinders and 
the circles are pitch circles, They are imag-inar}^ 
surfaces and lines, but they are eminently necessary, 
since the construction of the entire g'ear depends 
upon them. The point p is the pitch point, b^ng- 
alwa3^s upon //, the line oi centers. The distance />zt' 
or pzv is the circular pitch. It is the distance from 
a point on the pitch circle on one tooth to a point 
similarl}^ situated on the adjacent tooth. Formerl}'' 
this was universall3^ used in giving" the size of the 
teeth of gears. At present it has almost gone out 
of use, having been superseded by diametral pitch. 
Diametral pitch is a ratio. It is the quotient obtained 
by dividing the number of teeth in the gear by the 
diameter of the pitch circle of the gear in inches. 
The following holds true for the numerical values 
of the two. On an}^ gear, the product of the diam- 
etral pitch by the circular pitch equals 3.1416. 
Hence — 

Diametral pitch 1 is equivalent to circular pitch 
3.1416". 

Diametral pitch 2 is equivalent to circular pitch 
1.5708". 

Diametral pitch 4 is equivalent to circular pitch 
0.7854", etc. 

Also circular pitch l" is equivalent to diametral 
pitch 3.1416. 

Also circular pitch 2" is equivalent to diametral 
pitch 1.5708. 

Also circular pitch /^" is equivalent to diametral 
pitch 6.2832, etc. 



TKKTH OF GEARS. 29 

The leng-th of the tooth is limited b}^ a circle qi 
called the addendum or point linCf and by the circle 
vt'i called the dedendum or root line. Addendum and 
dedendum are the distances from the pitch circle to 
the addendum and dedendum line respectivel3^ The 
dedendum is made somewhat larg-er than the adden- 
dum, and the difference is called the clearance. When 
circular pitch is used the addendum is 0.3 of the 
pitch and the dedendum 0.4, leaving- the clearance 
0.1, or one-third of the addendum. When diametral 
pitch is used the addendum is the reciprocal of the 
pitch in inches, and the dedendum —^ the addendum, 
leaving- the clearance }{ the addendum. The line 
op is the face of the tooth and pk is the flank, O2 is 
the point and 4k the root The width ^j in cut gears 
is made one-half of the circular pitch, and it is equal 
to the space pz. In cast gfears the tooth is slightly 
less than the space and the difference, shown at yz, 
is the side clearance or back-lash. This is usually 
made equal to about -25- of the circular pitch. The 
following- table gives in a convenient form the usual 
dimensions: 

DIAMETRAL PITCH = P. CIRCULAR PITCH = p. 

1 3 

Addendum ^- or -r^ p. 



Dedendum -| — p- or ^^ p. 

^. 111 
Clearance — w or -ttt p. 



4 p ^^ 10 

1 

p ^^ 10 



Height of Tooth -^ — 5- or ^ p 



30 KLKMKNTAKY MECHANICAL DKAWING. 

Thickness of Tooth -y^^- or jm P- 

(^ 8^+1 52 

Space -y^^ or j^^ p. 

Backlash -^ — p^ or ^^ p. 

Pace of Gear 8 pr or 2 -^ p to 3 p. 

For the Involute S\^stem the heio-lit of tooth is 

1 ^ 

frequentl}^ taken as 2 -^- or -y- p. 

If in Fig-. 24 we have our two c_vlinclers a and /; 
and we consider a section of each normal to the axis, 
we ma}^ consider these sections to be b}^ planes which 
are co-incident but which move independently. The 
point of contact of the circles is at p. If, now, we 
place two other circles, c and d, tang-ent to the first 
two at ^, and consider that all centers are fixed, we 
have the necessary arrang-ement to obtain the re- 
quired form for our tooth. For if we take the points 
of c and d which are now at p for tracing- points, 
and set the circles in motion so that the}^ roll to- 
g-ether without slipping- and a is turning- clock-wise, 
then the tracing- point on r will mark on the moving- 
plane of a the curve pe and simultaneously on /; the 
curve p/\ If we turn a counter-clockv/ise d will 
trace on a the curve p£', and on /; the curve p/^. 
From the method of g-eneration it is evident that as 
a and /; roll tog'ether p£' and p/i will always have a 
point in common, as will pc and pj\ If, then, we 
take for our tooth curve on a the line ^pe and on /; 
the line hpf'xi is manifest that they will g-ive a uni- 



TKKTH OF GEARS. 31 

form velocit}^ ratio between a and b. This solution 
is known as the C3xloidal solution, since the tooth 
curves are all forms of the c^xloid, epi-C3xloid, or 
hypo-c,vcloid. i>g and ff are epi-C3xloids, fh and 
^e are h)^po-C3xloids, while if either a ox b became 
infinite in radius, so that the circle became a rig-ht 
line, the curves g-enerated upon it wxuld both be 
C3xloids. The circles a and b are called the pitch 
circles, and their radii determine the velocit3^ ratio 
of their axes a and b, The3^ are alwa3^s tang-ent at 
f, the pitch point. The circles c and d are called 
rolling or generating circles', the3^ are also called the 
lines of action, since the point of contact between 
gpe and h;pfi^ alwa3^s on c or d, The3^ may be of 
an3^ size and are not necessaril3^ of the same size, 
but practicallv the3^ must lie within certain limits. 
The controlling- condition is this, that any two tooth 
curves which are to work together must be generated 
by the same line of action. For any pair of gears 
which are always to work together the rolling cir- 
cles should be comparativel3^ large in order to de- 
crease the loss b3^ friction in the action of the teeth 
and to bring a larger number of teeth in action at 
once. The largest rolling circle that is commonl3^ 
vised is one whose diameter is equal to the radius of 
the pitch circle within which it rolls. The h3^po- 
C3xloid generated will be a diameter of the pitch 
circle, and the teeth will have what is known as 
radial flanks. As the size of the rolling circle is 
decreased, the tooth becomes broader at the base 



EI.KMKNTARY MECHANICAL DRAWING. 



and narrower at the point, consequent!}^ strong'er, 
but the frictional loss in action is increased and a 
smaller number of teeth is in action at a g-iven time. 
In Fig". 25 the point of contact moves along- the 
curve g-dspyf \N\iQr\ cd drives ah to the right. It 
cannot come upon a tooth of ah until it crosses the 
addendum or point circle st at 6 and it runs off ihe 
end of a tooth of (f<^ where it crosses the point circle 
qr at 7. The distance d-7 is the arc of action, and 
evidentl}^ increases as the radii of the rolling- circles 
^y'and^// are enlarg-ed. The portion 6p is the arc 
0/ approach and ^7 is the arc of recession, 

When a number of g-ears are to be constructed so 
that au}^ two of them ma)^ work together the_v are 
said to be interchangeable. In this case the same 
rolling circle must 
be used through- 
out; for if the 
g-ears A, B, and C 
in Fig-. 26 are to 
work interchang-e- 
abh% then must A 
work with B and 
C, and B must work 
with C. Then, of the curve 7npji on A, the portion 
p7i is the epic3xloid g-enerated b}^ E when A works 
with B, and b}^ F when A works with C. Hence E 
and F must be of the same size. Similarl}^ of the 
curve op' g the portion p' g must be g-enerated b}^ D 
when B and A are working- tog-ether, and by F when 




Fig. 26. 



TKKTH OF GKAKS. 33 

B and C are in mesh. Hence D and F must be of 
the same size and it is evident that the same rolling- 
circle must be used throughout for a set of inter- 
changeable gears. 

The smallest gear that can be used in this S3^s- 
tem is given b3^ some authors as one of twelve teeth, 
and b}^ others as one of thirteen, but the former is 
more commonly used. Hence for au}^ set of inter- 
changeable g-ears the rolling circle is taken equal to 
one-half the size of the smallest pitch circle of the 
set, and if a set is required which may be extended 
to include gears of an}^ number of teeth the smallest 
g^ear in the set is considered to have twelve teeth, 
the rolling circle is taken Vo the size of this pitch 
circle, and this method gives what is known as the 
general soltttion. 

To construct the drawings for a pair of spur 
gears, then, the following is the usual process. 
The velocity ratio and the distance between the 
axes being known, find the diameters of the pitch 
circles from the following equation: — 

ri = — r— - d, and r., = — r— - d; 
a + b ^ a + b 

in which r^ and r.^ are radii of the pitch circles, d is 
the distance between the axes, and a\ b is the veloc- 
ity ratio. 

Draw^ the line of centers and lay off upon it at 
the proper distances the centers and the pitch point. 
Draw the pitch circles. Determine the pitch to be 



34 ELEMENTARY MECHANICAL DRAWING. 

used, and from the pitch and pitch circles find the 
number of teeth upon each wheel; divide each pitch 
circle into as many equal parts as there are teeth in 
the wheel, beg-inning- for convenience at the pitch 
point. From these points of division la}^ off each 
tooth, taking- notice that the tooth of one wheel 
must come opposite the space of the other. From 
the table of proportions determine the addendum 
and dedendum, and strike in the point and root cir- 
cles. Assume the rolling- circles that are to be used. 
Construct the curves for faces and flanks of the 
teeth. Place a sheet of transparent celluloid over 
the drawing-, and stick a fine needle throug-h it into 
the center of the pitch circle. With a needle point 
trace upon this sheet the tooth curve on the wheel 
to whose center it is fastened, extending- this curve 
some distance be3^ond the point and root circles. 
Remove the celluloid and with a keen knife cut out 
the curve, thus making- a templet. Pin it ag-ain to 
the center of the pitch circle, and with the templet 
strike in the face curve on one side of each tooth of 
the wheel. Turn the templet over, and in the same 
manner strike in the other face of each tooth. Each 
wheel is constructed in the same manner. In cut- 
ting- the templet care should be taken that the edg-es 
are left smooth and that it exactl}" fits the curve 
from which it is taken. An error commonh^ made 
by beg-inners is at the point where the curve crosses 
the pitch circle. At this point both portions of the 
curve come tang-ent to the radius; this is an import- 



TEETH OF GEAES. 35 

ant point that should not be overlooked. When the 
teeth are constructed the rim, arms, and nave are 

Q 

drawn. The face of the g-ear is usually about ^ 

for diametral pitch, or 2/^2 ^ to 3^ for circular pitch. 
Where there is room on the shaft the nave is made 
from 25% to 50% g-reater than this in depth, that 
is, in the direction of the axis. The size of the ke}^ 
is g-iven b^^ Unwin empirically as follows: — 

b = 54: d + >^"; t -= .^ b; 

where d — diameter of shaft, b — breadth of key, 
/ = the mean thickness of the ke3\ In fitting this, 
% of the thickness is cut in the nave and ^'^ in the 
shaft. Unwin says that the thickness of the metal 
of the nave should be 

in which p is the circular pitch, and r the radius of 
the wheel; and the length should be at least three 
times the thickness. It is more common in practice 
to roughly proportion the nave to the diameter of 
the shaft upon which the wheel is fixed. If rf is 
the diameter of the shaft the boss, or enlarged por- 
tion of the shaft for receiving the wheel would be 
1.17 d, the nave thickness Yz d and the nave length 
1.63 d. The older forms of arms were commonl}^ a 
cross or a T, but at present the best gears have 
arms of elliptical or segmental cross-section, the 
longer axis being in the plane of the wheel. In the 



36 



KI.EMKNTAKY MECHANICAL DRAWING. 





Fig. 27. 



seg-mental section, Fig-. 27, the thickness is one- 
half the breadth, in the elliptical section it is four- 
tenths. The number of 
armsislarg-eb^arbitrar}^ 
and the calculation of 
their cross-section ac- 
cording- to formulse 
given b}^ Reuleaux and 

Unwin do not give satisfactory results. At the 
nave the breadth should be about thirty per cent, 
greater than at the rim. 

In the best practice the rim 
is made of the form shown in 
Fig-. 28. The dimensions shown 
are in terms of the circular 
pitch. 

THE INVOLUTE SOLUTION. 

In Fig-. 29 let ah and erf be 
two pitch circles, having- their point of tangency at 
p and their line of centers //. Draw a line r5 mak- 
ing a conven- 
ient angle 
with //, and 
from the cen- 
ters of the 
pitch circles 
let fall per- 
pendiculars 
upon rs at k fig. 29. 





THETH OF GEARS. 37 

and /. Draw the circles ^y and gh throug-h k and /. 
Now, if ^y rotates with ab and gh with cd it is evi- 
dent that the linear velocit}^ of a point on ^yis equal 
to that of one on gh, and the line rs will roll upon 
^y'and^7/ without sliding-. If a point of rs be taken 
as a tracing* point it will describe upon the plane of 
ab the involute of the circle ef 3.nd upon the plane 
of cd the involute of gh. Since these curves are 
g-enerated simultaneous!}^ b}^ the motion of a single 
point the}" have always a point in common, and 
since ab and <^<^ are rolled tog-ether to give the re- 
quired motion for g-enerating- the curve, these invo- 
lutes ma}^ be used as tooth curves to g-ive a uniform 
ang-ular velocit}" ratio. The line of action is rs, and 
for cut g-ears is taken so that it makes with the tan- 
g-ent to ab and cd at p an angle whose sine is 0.25, 
or 14° 27'. For cast g-ears it is taken so that this 
angle is 15°. ^/'and gh are the base circles or gen^ 
crating circles, 

To design an involute g'ear, Fig". 30, draw the 
pitch circle and divide it as in the cycloidal method. 
Draw the line of action throug-h the pitch point 
making- an ang-le of 75° with the radius and let fall 
upon it from the center of the g'ear a perpendicular. 
Describe a circle upon this perpendicular as radius. 
This is the g-enerating- circle, and from it is gener- 
ated the involute which is the tooth curve. Since 
this curve comes down to a cusp at the point where 
it touches the g-enerating- circle, no contact between 
the teeth is possible below this line. However, for 



38 



EI.KMKNTARY MECHANICAL DRAWING. 



the sake of clearance it becomes necessar}^ to cut 
the space lower than this line, and the side of the 




y 



Fig. 30. 



tooth is made radial below the g-enerating- circle. 
It must not be forgotten that there is no contact 
upon a tooth below the base circle but that this 
extra depth is mereh" for clearance. 

In his Odontics Grant makes the following- state- 
ment in regard to the length of tooth permissible 
upon the involute g-ear. W/ie7i two gears are in 
action^ the teeth of one cannot sweep over the point at 
which the line of action is tangent to the base circle of 
the other. If the tooth is extended be3^ond this point 
interference will occur, and the wheels will not work 



TEETH OF GEARS. 



39 




at all, or else the points on one will undercut the 
flanks on the other. Fig-. 31 shows this effect, 
where the rack tooth has under- 
cut the tooth of the pinion. 
Where the wheels are large this 
limitation does not affect the 
length of the tooth, but on 
small gears it is often the con- 
trolling condition in determin- 
ing it. Where it is possible, 
the tooth extends above and below the pitch circle 
exacth^ as in the C3x]oidal S3^stem; that is, the ad- 
dendum is— and the dedendum 25% greater. The 

involute rack tooth takes a special form. The tooth 
curve becomes a right line perpendicular to the line 
of action. This is the simplest possible form of the 
tooth curve, and is shown in Fig-. 32. 



Fig. 31. 




Fig. 32. 



THE BEVEL GEAR. 

The foregoing forms of teeth are used onl}- when 
the axes of the gears are parallel. When the}^ are 
not parallel, they ma}^ lie in the same plane or not. 
In case thev are in the same plane they will inter- 



40 KTvKMENTARY MECHANICAL DK AWING. 

sect, and ma^^ be connected b}^ the ordinar}' bevel 
g-earing-. If the_v are not in the same plane they 
cannot intersect, and ma}^ either be connected b}- 
an intermediate shaft and two pairs of bevel g-ears, 
or the}" ma}" be directly connected b}^ spiral gears, 
worm g'ears, or spiral bevel g-ears. The simplest 
case is that in which the axes intersect. The nor- 
mal surface, which before was a plane, is here a 
cone having- the same axis as the pitch cone, and 
g-enerated b}- a line which is perpendicular to the 
g-enerating- line of the pitch cone. Thus, in Fig. 33 
if ab and ac are the axes intersecting- at a, and <://' is 
the element along- which the pitch cones are tan- 
g-ent, then ^2/ revolved about ac generates the pitch 
cone, and ^'perpendicular to af\ revolved about ac 
generates the normal cone. So also (7/'and /// gen- 
erate the pitch and normal cones on the axis ah. 
The teeth are cut off at the smaller end b}- similar 
cones g-enerated by mk and Ik revolving: about ab 
and ^/<: respectiveh". 

To construct the drawing-s for a pair of bevel 
g-ears it is necessary to show the projections of the 
g-ears upon a plane parallel to their axis, a projec- 
tion of each gear upon a plane perpendicular to its 
axis, and the development of the intersection of the 
teeth with the normal cones. The quantities usuall}" 
g-iven are the angle between the axes, the pitch, and 
the number of teeth upon each gear, from which the 
pitch diameters can be found. The pitch is meas- 
ured at the intersection of the pitch cone with the 



TEKTH OF GEARS 



41 



normal cone, and on the larg-er ends of the teeth. 
If we have g-iven ah and ac as axes of a pair of bevel 
g-ears intersecting- at a, with radii ^'and ej\ having 
24 and 18 teeth of diametral pitch 4, ae. is laid off 
equal to df and ad equal to ef, and df-A.nd ^/ drawn 
to form the parallelogram ^^(i/e:'. The diagonal af 
of this parallelogram is the element of tangenc}^ of 
the pitch cones, and gh, drawn through / perpen- 



w ^ 




Fig. 33 



42 KLKMKNTAKY MbXHANICAL DRAWING. 

dicular to at\ g-ives the elements ^y'and fh which 
g-enerate the normal cones. The length of the teeth 

which is equal to — is laid off from /toward a^ g'iv- 

ing- k, and throug"h this point ;;// is drawn parallel 
to gh, g'iving' the elements mk and Ik of the inside 
normal cones. Assume uv perpendicular to ab and 
project y and k upon it g^iving-/'' and k'\ and from 
a" as center strike circles throug'h these points. 
These are the pitch circles for the outside and inside 
ends of the teeth on the smaller g-ear. In a similar 
manner assume wx perpendicular to ac^ project /' 
and h upon it at /' and k' , and from a as center 
strike in the pitch circles for the larg-er g-ear. 

To develop the normal cone, — From g and // as 
centers strike the pitch circles fj and />', and upon 
these pitch circles construct a pair of ordinar}^ spur 
gears as shown in the shaded figure. These are the 
developments of the outside ends of the teeth. In 
a similar manner construct a pair of spur g-ears 
upon the radii mk and //% transferring- this line to 
the position /;/i l^ to avoid confusing- the drawing. 
The same method of construction must be used as 
in the outer cones. The form kqr thus found is the 
development of the inside ends of the teeth. 

To construct the projections of the gear upon the 
planes normal to the axes,— The teeth on // are the 
development of the outer ends on the smaller gear. 
Carry the point and root circles in this development 
to the intersection with g/\ and project the points 



TKKTH OF GEARS. 43 

thus found upon nv at / and 2, Throug-li / and 2 
strike circles about the center <:/'. These are the 
point and root circles in the normal projection, and 
since the projection is parallel to the plane of the 
circles they are in their true si/.e. Divide the pitch 
. circle y"'^";/" into a number of parts equal to the 
number of teeth and throug'h these points of divi- 
sion drav/ the center lines of the teeth to a' , On 
these center lines la}^ out the form of teeth, taking* 
the width on the point, pitch, and root circles from 
the development //. In a similar manner construct 
the projection of the inside ends of the teeth, work- 
ing- from the development kq. Connect the points 
of the outer ends with the corresponding- points on 
the inner ends of the teeth by rig-ht lines which 
must, if extended, pass through a" . The construc- 
tion for the projection y d f is similar to that of 
f" e" n\ the width of the teeth and the size of the 
point and root circles being- found from the develop- 
ment,/;' and >^r. 

To construct the projection upon a plane parallel 
to the axes, — The projections of the axes are ab and ac, 
Throug-h the intersection of the point and root cir- 
cles on yy with g-Ji draw rig-ht lines perpendicular to 
ab^ and through the corresponding- point on //draw 
rig-ht lines perpendicular to ac. The projection of 
any tooth, as op, o' p\ is found by projecting the 
points from the point, pitch, and root circles at o' p' to 
the corresponding- lines at op. In this projection all 
elements of the conical surfaces must pass through a. 



44 



ELEMENTARY MECHANICAL DRAWING. 



THE SCREW, 



ir a plane li*>*ure, such as the quadrilateral rinos 
in Fig*. 34 be g-iven a uniform motion of rotation 
about an axis in its plane f\ 

and at the same time a 
uniform motion of trans- 
lation in the direction of 
that axis, the figure will 
describe in space that solid 
which is known as a screw 
thread. From the nature 
of the motion it is evident 
that each point in the 
plane figure generates a 
helix, and that the pitch 
of all these helices is the 
same, it being- the distance 
which each point moves 
along the axis while the 
plane rotates through one 
complete revolution, or 360°. The pitch of these 
helices is also the pitch of the screw. The point d 
in making one complete revolution about the axis 
AB passes through e to the point c, therefore cd is 
the pitch of the screw. If this is the same as the 
distance mn or md between two adjacent threads 
the screw has a single thread, if it is twice as g-reat 
as ;//;/ it is a double thread screw, and if three times 
as great, a triple thread screw. 




TEETH OF GEARS. 45 

The most common screw is the sing-le thread, 
but double and triple threads are not uncommon. 

The form of the g'enerating- iig-ure rinos g'ives the 
name to the thread. If it be a square it is a square 
thread, and the sides of the square will be to the 
pitch in the ratio of 1: an even number, as 1:2, 1:4, 
1:6. In the sing"le square thread screw the side of 
the square is one-half the pitch. The square thread 
is used to a considerable extend in screws that re- 
ceive a ^reat deal of wear, since it wears better and 
works with less friction than does the triang-ular 
thread, althoug-h, on the other hand it is not so 
strong*. 

The most common form of thread is the trian- 
g-ular. In Kngland the Vv hitworth triangular thread 
is used and in America the Seller's or U. S. Stand- 
ard. In the Seller's thread the lines rs and mo each 
make an ang-le of 60° with the axis, and the lines 
)'7n and sit are right lines parallel to the axis and 
equal in length to one-eighth /;///. In the Whit- 
worth thread the lines I's and mn inclose an angle 
of 55° and the right lines inr and us are replaced b}^ 
circular arcs. 

To make the drawings for a screw assume the 
axis AB and upon it a center O. from which strike 
the circle f'^q which is the plan or end vieYv^ 
Through ^ draw ^c parallel to the axis and lay off 
upon it spaces equal to the heig-ht of the, thread, 
mil. Lay off below each of these points a distance 
equal to one-eighth of mn, and draw the lines cor- 



46 



KI^BMBNTARY MECHANICAL DRAWING. 



responding- to 7no and rs. Prom the intersection of 
rs and mi Vd^Y off toward ;/ a distance equal to one- 
eig-hth 71111 g-iving 21^ and throug-h this point draw 
ov parallel to AB. Draw the circle i.'w and the 
lines wh and qe parallel to AB. If the screw is to 
have a sing-le or triple thread, project the points 
ixom. i)C io wh and from vo to qe; if a double or 
quadruple thread, project the points from pc to qe 
and from ov to zvh. 

Construct a helix of amplitude pq and of pitch 
equal to that of the screw, and with it connect the 
points on i>c with those on qe. Construct a second 
helix of amplitude vw and of the same pitch as the 
first and with it connect the points on ^z' with those 
on wh. On the top side of the thread draw the line 
tk tangent to the two helices at / and k, and on the 
lower side draw t' k' in a similar manner. The curve 
from e to /is similar to that at t' k' . The onh^ line 
of the orig-inal section rmos which remains visible 
is the line rni and a small portion of sti. 

The standard sizes and numbers of threads as 
adopted b}^ the U. S. Navv are given in the follow- 
ing table. This is the standard now universal!}^ 
used in the U. S. The letters refer to the dimen- 
sions shown in Fie:. 35. 



T 

CI 

k 



n T 



© 



Fig. 35. 



TEETH OF GEARS. 47 



a No. of threads b c e d 

-^20 -T -?- 0.185 4^ 

4 18 1-4 0.24 -f 

1 16 -i- 4 0.294 4^- 

-^14 ^ ^ 0344 ^ 

T" 1-^ Tft" ~X 0.4 -^ 

— 12 — — 454 ~ 

-|" 11 -32". ^x 0.507 1^ 

3 ,^ 5 3,^,^ ,1 
-4" 



10 -8" ^ 0.62 l-i" 



8 " 32 8 0.731 1 j^ 

18 -rl" 1 0.837 l-f- 

1^ 7 ^ 1-i 0.94 1^ 

l4- 7 1 1^ 1.065 2 

1^ 6 1^ 1^ 1.16 2-^ 

14-6 lif 14 1-284 2^- 

l4 54- 14- 14 1.389 24 

3 , ,„, ^ 3 

^ 4 



1^ 5 l-i" 1-^ 1.491 



l-T 5 1— l4" 1-616 2-i|- 



j^ 



2 44- 14 2 1.712 34- 

24 44- 14 24 1-962 34- 



48 ELEMRNTAKY MECHANICAL DKAWINO. 



a No. of threads /> c e d 

2-4- 4 l4f- 2-4- 2.176 3 ~ 

2^ 4 2-^ 2^ 2.426 4 | 

3 3 4" 2 -J- 3 2.629 4-4 

3-f 3^ 2-4- 3-4 2.879 5 



3 4- 3^ 2^ 3-!r 3.100 



3—4- 3 2 — g- 3—4- 3.317 



o 



4 - 16 ^2 -^.-1"^^ -' x 



D 



O 



4 



4 3 3^ 4 3.567 6 4" 
44- 2-4 34- 4-4 3.798 64 
44- 24- 34- 4-4 4.128 64" 
4-4 24 3-4 44 4.256 74 

5 24 34 5 4.480 74 
54 24 4 54 4.730 8 
54 24 44 54 4.953 84 
54 24 44 54 5.203 8 4 



DIRECTIONS FOR PLATES IN DRAWING I 

GENERAL. 
All plates in this course shall be of the size 

12 X 15 . 

Each plate shall have a border line removed /4" 
from the edg-e all around; thus making- the border 
rectangle 11" x 14". Width of border line, -^" ^ 

In the upper right hand corner of each- plate, 
and -^" from the upper border line shall be placed 
the j)Iate ntmiher in vipright capital letters }i" high, 
thus; 

PLATE 23. 



theletterP being 1/^" from the corner of the border 
rectangle. 

All plates, excepting those traced from given 
copies, shall bear the name of the draftsman at the 
lower right hand corner in capital letters j'i" high 
and -j^" belov/ the lower border line, the name to 
begin 2" from the right end of border line. 

Similarl}' the date of the completion of the plate 
shall be placed at the lower left hand corner, begin- 
niner %" from the left end of border. The date 



50 KLl^MKNTAKY MECHANICAL DRAWING. 

shall be of the form 1905-4-26, in the order, year^ 
mouthy day. 

Unless otherwise instructed, each plate shall 
lirst be done /;/ pencil and submitted to the in- 
tructor for approv^al or corrections. If approved, 
he will place the word ''/;//('" on the marg-in. When 
the plate is linished, it shall again be submitted to 
the instructor before the word ^' Ink''' is cut off; and 
if approved he will place his signature upon the 
plate, which shall then be trimmed to the proper 
size and handed in. 

Do not submit an}^ pencilled drawing* for ap- 
proval until the preceeding plate has been handed in. 

One plate from each set will be retained as the 
property of the University. 

Pirate 1. (upright letters). 

Secure a sheet of the ruled paper io the board, so 
that the upper edg-e is parallel to the upper edge of 
the T square blade. Dravv^ the border rectang'le so 
that the rig*ht and left ends are equall}^ distant from 
the ends of the blue lines (they are nearly, but not 
exactl}, 13'' long-). Draw a light pencil line through 
the muldle of the plate, perpendicular to the blue 
lines/ Pill the space occupied by the blue lines, with 
a system of 22/4° lines about 2" apart, to be used as 



g-uide lines for the inclination of the lettersJ- 

Read pag-es 13-20 in Reinhardt's Lettering- Book. 
Note carefully the formation of each letter and 
numeral. 



DIRECTIONS FOR PI.ATKvS IN DRAWING. 51 

The plate consists of /^ row each of the letters 
and numerals in the order g:iven in the text-book, 
omitting" lower case /, and /, capital I, and numerals 
1 and 0, and adding- & (see pp, 21 and 35). 

Put the letters about /s" apart and begin each 
line at the extreme left end of the blue lines. 

Make about 5 letters of each kind with a 4H 
pencil, submit for approval, and do the others 
directh^ in ink, using Gillott's No. 303 pen and 
Waterproof India Ink. 

In order to avoid soiling the lettering plates, the 
space not in use should be kept covered by a cloth 
or sheet of clean paper; or the hand may rest upon 
a blotter, while lettering. 

PI.ATK 2. Cltpright text). 

Read pp. 20-22, Reinhardt. Beg-inning wnth the 
chapter on Capital Letters, p. 16. copy the text in 
upright letters until the plate is tilled. Place the 
caption in the middle of the first line. Indent all 
paragraphs f4". Observe instructions as to spacing. 
The letters and numerals are correctly shown in 
Reinhardt's Plate 1, lines 6, 7 and 8. 

Plate 3. (inclined letteks). 

Read pp. 5-13, Reinhardt. Make a half row 
each, of the uprig-ht letters and numerals, in the 
order given in the text-book, omiting /, /, I. 1 and (> 
as in Plate 1. , 



52 klembntaky mechanical dkawing- 

Plate 4. (inclined text). 

Use 22/^2° g-uide lines as in Plate 1. 

Cop.v caption and text beg-inning- with ''''Inclined 
Lettering,'' p. 5. The letters and numerals are cor- 
rectly shown in Reinhardt's Plate 1, lines 1, 2 and 3. 

Plate 5. (inclined text). 

Copy text beg-inning" with '' Lower Case Letters^''' 
p. 7, Reinhardt. 

Plate 6. (inclined text). 

Cop3^ text beginning with '' Niimerah,'' p. 11, 
Reinhardt. 

Plate 7. (numerals). 

Make ten lines of inclined numerals, 12 3 4 5 6 
7 8 9 &, placing seven groups on a line and separ- 
ating the groups by an extra space. Fill the next 
ten lines in a similar wa}^ with ?^^n^/^/ numerals. 
Fill the next live lines with incli?ied Roman numer- 
als I II III IV V VI VII VIII IX X, placing four 
groups on a line. 

Fill the last four lines in a similar way with 
lifright Roman numerals. 

Plate 8. (table). 

Use unruled paper. Copy one page of a five- 
place logarithmic table, placing it in a heavy border 
rectangle ll" x 8". Above the border of the table 
put: COMMON LOGARITHMS OF NUMBERS 
in inclined capital letters yi" in height. 



DIRECTIONS FOR PLATES IN DRAWING. 53 

Make the long'est dimension of the table corre- 
spond to the long'est dimension of the plate and 
arrang-e the whole exactl}-^ in the middle of the 
plate. 

Plate 9. (fractions). 

Use ruled paper. Make fractions -^^ -g-^ -^^ -j-^ 

-^^ -T^ lit 1' 1 1^^ 1-f ^ etc., increas- 
ing- the value of each fraction b}^ -jr-. See Rein- 
hardt's Plate 1. 

Plate 10. (titles). 

Read pag'es 26-31, Reinhardt. Reproduce the 
six titles of Plate IX, Reinhardt, double size^ on un- 
ruled paper. 

Draw two lig-ht pencil lines parallel to, and 3^^' 
from the left and rig-ht border lines. Make the 
titles S3'mmetrical with respect to these lines, and 
space the titles uniforml}^ with respect to upper and 
lower border lines and with respect to each other. 

The titles in the lettering- book are not exactl}^ 
S3^mmetrical. The student will correct this in his 
plate. 

Plate 11. (practice with compass and ruling 

pen). 

Read pages 3-6 of Notes. 

Use a 6H pencil for all mechanical drawings that 
are to be inked. The pencil must be kept sharp, 
and the lines made fine and lig-ht so that the exact 
point where two or more lines intersect can easil}" 
be distinguished. 



54 KLKMKNTAKY MECHANICAL DKAWING. 

Construct within the border rectan^jfle the out- 
lines shown in the ii.^ure (see blackboard sketch 
or blue-print) drawing* all straig-ht lines w^ith the 
T square head on the /e/Y edge of the drawing" board; 
vertical and oblique lines being* drawn with the 
triang-les fixed ag-ainst the upper edge of the 
T square blade. 

Take g-reat care that (I?) {c) and (d) are perfect 
three inch squares; otherwise the final results will 
be far from accurate. 

I^arl (a) consists of six groups of lines most fre- 
quently used in mechanical drawings. Each group 
consists of three lines J{" apart. Distance between 
g-roups, /4'\ The pencil lines may be drawm lig-ht 
and continuous. When inked in, the first line of 
each g-roup shall be /i£'/it, the second med/u?n, and 
the third heavy, as shown on the blue-prints. 

Group. 

1. —full line 

2. — — — ^ broken line 

3. — — — — — — dash-dotted line 

4 point-dotted line 

5. • • • broken and dotted 

6. lon^ and short 

dashes. 
Leng'th of longf dashes, aprox. y-i" 
'' '' short dashes, '' j^ " 

spaces, -^ 

Par/ ih) is an exercise for the T s(iuare and 



DIRECTIONS FOR PLATES IN DRAWrNTG. 55 

triaiig-les. Draw the diag'onals and diameters. 
From the left end of the horizontal diameter, ah^ 
lay oil distances ah — V2", and be — Vz" (See black- 
board sketch). La}^ off similar distances from the 
rig-ht end of the horizontal diameter. Throug-h a 
draw lines above and below making* 75"^, 60°, 45° and 
15° respectivel_Y with the horizontal diameter ah. 
Through h and c draw 60° and 15° lines. Prom the 
points g, d^ e and /', where these several lines cross 
the diag-onal, draw lines making- 15°, 30°, 45°, and 
75° with ah. Repeat this for the other three quad- 
rants. 

^V<9/^.— Whenever several lines converg-e to a 
point, as in this lig-ure, it is best, when feneilling, 
to leave a clear space around the point in order that 
its exact position ma}^ not be lost. The pencil lines 
need not be brought closer than -^" from the point. 
When inking', these lines will be broug^ht to the 
point if the drawing- requires it; but if the}^ are 
mereh^ auxiliar}^ lines or construction lines, the}^ 
should be term^inated about Y^" from the point. 

Part (e). Exercises in drawing- arcs tangent to 
rig-ht lines. Use T square, triang-les and bow-com- 
passes. Draw^ diag-onals and diameters. Divide a 
horizontal and vertical side into half inch spaces. 
Throug-h the points of division, with 45° triang-le, 
draw lines parallel to the diagonals. Through the 
points a, b, c, d, e, t\ g, //, etc., as centers, strike 
tangent arcs, as shown in the sketch. Repeat for 
the remaining- quadrants. The small arcs should 

L.OT U. 



56 EI.EMKNT.ARY MECHANICAL DKAWINO. 

all be drawn without chang-in^ the radius. Simi- 
larl)^ for the larg'e arcs. 

Part {d). Arcs tan«*ent to arcs. Use T square, 
triang-les and bow-compasses. La}^ off ab, ae, etc., 
equal to >^". Throug'h e, /;, etc., dravv^ with 
T square and triang-les, lines en, bk, etc., parallel 
to the sides of the square. These lines should in- 
tersect in the diameter at //. With k, h, ;/, etc., as 
centers and V^" radius, describe circles. With //, o, 
etc., as centers, and a radius equal to Y-z of oh de- 
scribe circles which are tang-ent to one another at j6. 
With k, h, n^ etc., as centers, and radius r;/, de- 
scribe the small circles. 

If accurac}^ is to be expected g'reat care must be 
taken in laying- off the required distances, and in 
taking centers at the exact points of intersection of 
the lines. The graphite in the compasses miist be 
hard and sharp. 

Full and dash-dotted lines are to be used as 
shown. Parts {b) (c) and (d) are to be inked in 
mediu7n weight lines. 

Part {e) consists of six concentrix circles j<(" 
apart, the largest having- a radius of 2". The lines 
are all medium weight, and corresponding- to the 
six forms of part (a) in order; the outer circle being- 
a full line. 

Part {/) consists of twelve concentrix circles in 
pairs. The space between any two lines is equal 
to the width of the heavier line. The larger spaces 
between the pairs are %" . The widths of the lines 



DIRKCTIONS FOR PLATES IN DRAWING. 57 

beg-inningf with the outer circle are heavy, lig'ht; 
medium, lig"ht; two lig'ht lines; two light lines 
ag-ain; light, medium; light, heav3\ 

Plate 12. (geometrical problkms). 

Fig, I. With center at (>^", 7"),— i. e., ^" 
from left border line, and 7" from /<9zt'<?r border line, 
— and radius 2/^", describe the upper right hand 
quadrant of a circle. With center at (5^", 10/^")-. 
describe an arc tangent to the first arc. Connect 
the two centers b}^ a right line, and at the point of 
tangenc}^ construct the common tangent, perpen- 
dicular to the line of centers. From the point of 
tangenc}^ lay off on the smaller circle an arc whose 
chord is 2/i". Lay this arc off first on a right line, 
and then on the larger arc, by the methods of Figs. 
1 {a) and 1 (^). 

Fig, 2 (a). Divide a line 3" long into 7 equal 
parts b}" the method of Fig. 2. Place the line par- 
allel to the upper border, its left end being at 
(6/i", 9/^")- (^). Divide a line 3" long into two 
parts proportional to the side and diagonal of a 
square, b}' method of Fig. 2. Place left end of line 
at(6^", 6>r'). 

Fig, J, Construct a regular decagon upon a 
side V in length. Show the circumscribed circle 
in a light broken line. Put left end of given side 
atdl'-', 6^"). 

Fig, 4, In a circle of lj4" radius, inscribe a 



58 EI.KMKNTAKY MECHANICAI, DKAWING. 

reofular pentag-on. (Fi^. 4). Center of circle at 

(2", SM'"). 

Fig", 5. Construct an ellipse with axes Syi" and 
3'', b}" the method of Fig". 7. Center of ellipse at 

ibVA.vA"). 

Fig\ 6. Inscribe ten circles of ~A" diameter in 
another circle. Center of lowest circle at (12" 2"). 

Note. — In all ^reometrical problems of this and 
succeeding- plates, make given lines heavy, (^^"); 
required lines inedium, and construction lines light, 
dash-dotted as in the preceeding- plate. Place FIG. 1, 
FIG. 2, etc., in a convenient place in capitals Y^" 
high. 

Construction lines, when drawn in pencil, should 
he full, not broken. 

Letter each figure as in the text-book. 

PivATK Ivl. (geometrical PROBLEMS). 

Fig-. 7. Construct an ellipse b}^ the trammel 
method. Axes 4" and 2/^". Center of ellipse at 
(2/'2'', S}i"). Construct two tangents from a point 
without at {AW\ 9/i"), and a tangent from a point 
on the ellipse iVs" from left border line. 

Fig-, 8. Construct the two nappes of an h)^per- 
bola. Distances between vertices 134^". Each focus 
is 1/2" from the corresponding vertex. Left hand 
focus at (6", Sy^" ), Construct two tang-ents to the 
left hand nappe from a point without, at (6^", 9") 
and a tangent from a point on the rig-ht hand nappe, 
IV^" from lower border. 



DIRECTIONS FOR PLATES IN DRAWING. 59 

Fig, g. Construct a parabola on a 3" base, the 
vertex being: 3/4^'' to the left of the base. 

Put vertex at (10", 8/^"). Construct two tan- 
g-ents from a point at (11", 6^/4"), and a tang-ent at 
a point on the curve, 9" from lower border line. 

Fig, 10. Construct an ellipse with horizontal 
axis 3/^", and vertical axis 5/^" long. Center of 
ellipse at (2'^", 3/^"). Construct the upper half 
by the method shown in the left half of Fig-. 5 {a). 
Construct the lower half b_v the method shown in 
the left half of Fig. 5 (h). 

Fig, II, Construct one complete convolution of 
a spiral of Archimedes, with origin at (6/^", 2%"). 
The curve passes through a point 3" to the right of 
the origin. Find sixteen points on the curve. 

Fig. 12, Construct one convolution of a logar- 
ithmic spiral with origin at (12/^", 2^"). The 
curve passes through a point 2/^" to the left of the 
origin. 

Plate 14. (roulettes). 

Fig, ij. Construct a c} cloid, with a tangent at 
some point. Radius of rolling circle = >^". Cen- 
ter of rolling circle at (3>^", 9"). 

Fig, 14, Construct an epicycloid with tangent. 
Radius of base circle = 3". Radius of rolling cir- 
cle = y%" , Center of base circle at (3/^", 2"). 

Fig. 15, Construct a h3^pocycloid with tangent. 
Radius of base circle =3"; Radius of rolling circle 
= yk" , Center of base circle at (3/^", 4"). 



(A) KLKMENTAKV MKCHANICAL DKAWING. 

fy^^'. i6. Starting with the highest point on a 
circle of l^'s" radius, construct an involute, carry- 
ing- it throuo-h ISif. Center of circle at (9", 8>^"). 

M£\ ly. Construct an internal epic^^cloid. Ra- 
dius of base circle, 1/8 ". Radius of rolling- circle, 
1 -3^". Center of base circle at (8/^", 3/^"). 

M^'. i6\ Construct the vertical and horizontal 
projections of one complete convolution of a helix. 
Diameter, 2^''. Curve rises 3'' for each convolu- 
tion. Center of horizontal projection at (12/^", 2''). 
Determine 16 points on the curve. 

Plate 15. (link shading). 

I^i[^\ ig is a shaded hexag-onal prism with hexa- 
g-onal abacus (Fig's. 19 and 20). Put the center of 
the auxiliar3^semi-circumferencewiiich circumscribes 
the half hexag-on at (2'\ 8"). The radius is l^X". 
The distance a a" is 1^". Parallel \o a a" and>2" 
from it, draw the left vertical line of the abacus. 
Complete the drawing of the abacus, making- it /^" 
thick. Leng-th of prism is 5/^". Determine the 
outline of the shadow. It will be convenient to 
draw^ the shade lines the following- distances apart. 

For the lig-ht surface, ^"> 
medium 



1 f, 



16 

dark 



20 

shadow -2^- . 

Do not put shade lines in pencil 



DIRECTIONS FOR PLATES IN DRAWING. 61 

Fig, 20 is a shaded cylinder with C34indrical 
abacus. (Fig-. 21) Center of semi-circumference is 
at (2", 3"). Diameter of abacus is 3/^". Diam- 
eter of C3^1inder is 2/^". Thickness of abacus is j^". 
Length of c_vlinder is 5/^'\ 

Do not put shade lines in pencil. 

Fig, 21, "Read the text on Rig-ht Line Hatching, 
p. 25. The m^ethods of ' 'hatching-" are shown in 
Fig-. 23. The upper left hand corner of the cast- 
iron rectang-le is at (9>2", 9M"). The six small 
rectang-les are each l'' x Va" and are separated 
from each other by /i". The stone rectangle is 
1/^" X \y\'\ It is placed directl_Y below the other 
six, and separated b}^ ^2". 

Fig, 22 is a sphere shaded by circular arcs. The 
radius of the sphere is IM", the center being- at 
(11>^", 2X2"). SeeFi^. 22. 

Do not draw the circles for shading in pencil, 
but indicate the points along one of the 45° lines, 
throug-h which the circles are to be drawn. Great 
care must be taken not to wear a large hole in the 
paper at the center of the circle. 

Pi. ATE 16. (tracing). 

Read pp. 22-26, Reinhardt. Trace the plate 
found in the envelope and entitled ''Slating- Ts and 
Ls." 

Stretch a sheet of tracing cloth over the plate 
with dull side up. To make the cloth take the ink 
evenly, sprinkle powdered chalk, talcum or mag-ne- 



62 ki.emp:ntaky mechanical dkawino. 

slum upon the surface, and rub with a clean dry 
cloth. 

In tracing", each fig-ure should be completed be- 
fore g'oing- to the next; for if the draftsman is 
unable to complete the plate at one sitting-, he will 
find that the cloth has shrunk during" the nig-ht, and 
that the lines drawn upon the cloth do not exactly 
cover the lines of the original. 

Whenever open holes are indicated by black cir- 
cles, the outlines of these circles should first be 
drawn with the bow compasses and then filled in 
with the free-hand pen. 

Put t\i^ opiate number in the usual place; but the 
name and date in the rectangle at the lower rig-ht 
hand corner of the plate. 

The tracings required to be done by the student 
in this course are intended to g-ive him an idea as 
to how mechanical drawing's are made and dimen- 
sioned. To this end the student should carefully 
stud}^ each drawing until he is able to explain it 
when called upon to do so. 

Plate 17. (tracing). 

Tracing of 'Xathe Apron Casting,'' on smooth 
side of cloth. 

Plate 18. f tracing). 

Tracing of ''Split Sleeve and Collar/' on dull 
side. 



directions for platks in drawing. 63 

Plate 19. (tracing). 

Tracing- of ''Pedestal Bearing-,'' on gloss}^ side. 
A full sized model of this ma}^ be found in Room 
418. 

Pirate 20. (tracing). 

Tracing- of ''Beams and Connections." Use dull 
side of clotli. 

Plate 21. (trx\cing). 

Tracing of '^Bottom Chord Joint" (gloss}^ side). 
A full sized model of this ma3^ be found in Room 
415. 

Plate 22. (tracing). 

Tracing- of ''Column" (dull side). 

Plate 23. ( cycloid al cut gears). 

Read pages 26 to 36 in notes. A c3xloidal cut 
spur g-ear of 12 teeth and 8'' diameter meshes on 
the /^y/ with an annular gear, diameter 22"; and on 
the right with a rack. 

Put radial flanks on pinion, and use the same 
rolling circle throughout. 

The thickness of the rim should be approximateh^ 
one-half the circular pitch. 

Show 6 teeth on annular gear, and 5 on rack. 
Put center of pinion at (7", 6"). Show all construc- 
tions and all important dimensions. Construct full 
size. In a rectangle at lower right hand corner, 
place title, as in tracings. 



64 EI.KMKNTAKY MKCHANICAT. DRAWING. 

In some other convenient place, put. 

No. of teeth on Annular Gear, 

Pinion, 

Dia. Pitch, 

Vel. Ratio, 

Plate 24. (invoi^utk cut gears). 

Read pag-es jd /6> ^^p of the notes. An involute 
cut spur g-ear of 20 teeth and 16" diameter meshes 
with a pinion, and the pinion with a rack on the 
rig-ht. Ang-ular velocit,v ratio, 3:4. Draw to scale 

/2" = r\ 

Show onh^ half of the larg-er wheel, its center 
being- at (2/i'\ 6"). Larg-er wheel has four arms; 
smaller one has none. Breadth of arm, IVa." . 

At the left, show a section through the axis of 
the larg-er wheel. Face, 2"; Nave, 2'^". Thick- 
ness of rim, 1^". Dia. of shaft, 134". Boss, 2". 
Thickness of nave, Vs'', 

Give all important dimensions, and arrange title 
and notes as in Plate 23. 

iVote. — Cut off teeth at interference point as ex- 
plained on page 38, and make heig-ht of teeth -{r 
as nearh^ as possible. 

Plate 25. (invoi.ute bevel geaks). 

Read pag-es 39-43 of the notes. Construct a pair 
of involute bevel g*ears. Angle between axes 90°. 
Dia. pitch -^. Teeth 15 and 21. 

Show the projections of a /ici/J' of the smaller 



DIRECTIONS FOR PI.ATKS IN DRAWING. 65 

g-ear below, and a little more than a quadrant of the 
larg-er g-ear to the rig-ht. Scale, /^" = l". Place 
intersection of axes at (4'\ 9/^"). 

Make title and notes according- to taste. 

Pirate 26. (sckews and bolts). 

Read pag-es 44-46 of the text. Construct: 

(t/). A U. S. Standard screw thread cut on a 
bar 6" long- and 2^" diameter. Thread double smd 
rig-ht-handed. Pitch 2". Show section of stud as 
in Fig. 34. 

(//). A similar stud with square thread. 

((f). A U. S. Standard bolt with hexag-onal head, 
and nut on bolt. Dia. l". Leng-th under head, 4". 
Sing-le, right handed thread. See table for other 
dimensions. Show threads b}^ straight lines instead 
of helical curves. Show top view of nut removed 
from bolt. 

(rf). A left handed bolt and nut (U. S. Stand- 
ard), showing threads b}^ heav}^ and light lines. 
Dia. yi" , Length under head 4", Omit top view 
of nut in this fig-ure. 



«'..\R. « 



U'w'J. 



